The conflict in the story Bud, Not Buddy is Bud's situation after his mother dies. Ten-year-old Bud Caldwell is left in the care of the state. He is placed in an orphanage and in a foster home with an abusive family. He runs away to find the man he believes is his father, a band leader whose name appeared on some flyers Bud's mother left him.
Remember that there are two major types of conflict: internal and external.
To identify the conflict of a story you have to ask yourself about the major problem. Is the problem within a character? Is it a moral issue, or a confusion that the character has about some aspect of life? If so, then it is an internal conflict within the character.
If you identify the problem as something that exists between two characters, or between the character and his or her environment, then the conflict is external.
In Buddy's case, he is conflicted within, about his father, and he also has some minor conflicts along the way with other characters and circumstances.